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Mirrors > Home > MPE Home > Th. List > issubc2 | Structured version Visualization version GIF version |
Description: Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
issubc.h | ⊢ 𝐻 = (Homf ‘𝐶) |
issubc.i | ⊢ 1 = (Id‘𝐶) |
issubc.o | ⊢ · = (comp‘𝐶) |
issubc.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
issubc2.a | ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) |
Ref | Expression |
---|---|
issubc2 | ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issubc.h | . 2 ⊢ 𝐻 = (Homf ‘𝐶) | |
2 | issubc.i | . 2 ⊢ 1 = (Id‘𝐶) | |
3 | issubc.o | . 2 ⊢ · = (comp‘𝐶) | |
4 | issubc.c | . 2 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
5 | issubc2.a | . . . . 5 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) | |
6 | fndm 6455 | . . . . 5 ⊢ (𝐽 Fn (𝑆 × 𝑆) → dom 𝐽 = (𝑆 × 𝑆)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐽 = (𝑆 × 𝑆)) |
8 | 7 | dmeqd 5774 | . . 3 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑆 × 𝑆)) |
9 | dmxpid 5800 | . . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
10 | 8, 9 | syl6req 2873 | . 2 ⊢ (𝜑 → 𝑆 = dom dom 𝐽) |
11 | 1, 2, 3, 4, 10 | issubc 17105 | 1 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 〈cop 4573 class class class wbr 5066 × cxp 5553 dom cdm 5555 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 compcco 16577 Catccat 16935 Idccid 16936 Homf chomf 16937 ⊆cat cssc 17077 Subcatcsubc 17079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-pm 8409 df-ixp 8462 df-ssc 17080 df-subc 17082 |
This theorem is referenced by: 0subcat 17108 catsubcat 17109 subcidcl 17114 subccocl 17115 issubc3 17119 fullsubc 17120 rnghmsubcsetc 44268 rhmsubcsetc 44314 rhmsubcrngc 44320 srhmsubc 44367 rhmsubc 44381 srhmsubcALTV 44385 rhmsubcALTV 44399 |
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